Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.

We investigate the problem of determining when $\IA (F_{n}({\bf A}_{m}{\bf A}))$
is finitely generated for all $n$ and $m$, with $n\geq 2$ and $m\neq 1$. If
$m$ is a nonsquare free integer then $\IA(F_{n}({\bf A}_{m}{\bf A}))$ is not
finitely generated for all $n$ and if $m$ is a square free integer then
$\IA(F_{n}({\bf A}_{m}{\bf A}))$ is finitely generated for all $n$, with
$n\neq 3$, and $\IA(F_{3}({\bf A}_{m}{\bf A}))$ is not finitely generated.
In case $m$ is square free, Bachmuth and Mochizuki claimed in ([7],
Problem 4) that $\TR({\bf A}_{m}{\bf A})$ is $1$ or $4$. We correct their
assertion by proving that $\TR({\bf A}_{m}{\bf A})=\infty $.